Is there a way to edit around another object?
As long as the meshes are different objects, editing one will not affect the other, in other words you can only edit one object at the time.

When you create an object in object mode it will be independent of other objects.

A more detailed explanation:

Blender uses a hierarchical structure to organize data.

You can Add a Mesh in Object Mode and Edit mode, but they are not the same. 

So what is the difference?

When you add an mesh in Object Mode you are creating both, a mesh and an object. Objects created in object mode are independent of each other, meaning that they will be named with a unique name and will all have their own separate origin (or pivot point), and can be transformed independently.

Meshes created in Edit Mode are part of the Active object, an already existing object that is being edited.

Objects can contain one or many meshes, connected or not.

All meshes within that object will share a common origin and will behave as what they are: a single object, meaning they will move, rotate and resize together when transformed in Object Mode.

For example: Here's a scene with four different objects created in object mode. Each one of them shows as a different object in the outliner.



If new meshes are added in edit mode, they become part of the active object, note how the outliner shows no other objects being created, all new meshes are part of the same object called "cube":